Method and device for performing transmissions of data

ABSTRACT

For determining, in a distributed fashion, precoders to be applied for performing transmissions of data between a plurality of transmitters and a plurality of receivers via a global MIMO channel H of a wireless communication system, said precoders jointly forming an overall precoder V, each and every j-th transmitter perform: gathering long-term statistics of CSIT errors incurred by each one of the transmitters; obtaining short-term CSIT related data and building therefrom its own view Ĥ (j)  of the global MIMO channel H; determining an estimate {tilde over (V)} (j)  of the overall precoder V from the short-term CSIT related data; refining the estimate {tilde over (V)} (j)  on the basis of the gathered long-term statistics of CSIT errors so as to obtain a refined precoder {tilde over (V)} (j)  that is a view of the overall precoder V from the standpoint of said j-th transmitter, further on the basis of its own view Ĥ (j)  of the global MIMO channel H, and further on the basis of a figure of merit representative of performance of said transmissions via the global MIMO channel H.

TECHNICAL FIELD

The present invention generally relates to determining, in a distributedfashion, precoders to be applied for transmitting data between aplurality of transmitters and a plurality of receivers in a MIMOchannel-based wireless communication system.

BACKGROUND ART

Wireless communication systems may rely on cooperation in order toimprove their performance with regard to their environment. According toone example, such cooperation can be found in a context of a MIMO(Multiple-Input Multiple-Output) channel-based communications network inwhich node devices, typically access points such as base stations oreNodeBs, cooperate in order to improve overall robustness ofcommunications via the MIMO channel.

So as to perform such cooperation, transmitters of a considered wirelesscommunication system rely on CSI (Channel State Information) relateddata and/or channel estimation related data for determining a precoderto be applied by said transmitters in order to improve performance oftransmissions via the MIMO channel from said transmitters to apredefined set of receivers. Such a precoder is typically determined ina central fashion, and parameters of the determined precoder are thenpropagated toward said transmitters for further applying said determinedprecoder during transmissions via the MIMO channel from saidtransmitters to said receivers.

It would be advantageous, in terms of system architecture and in termsof balance of processing resources usage, to provide a method forenabling determining the precoder parameters in a distributed fashionamong the transmitters. However, doing so, CSIT (CSI at Transmitter)mismatch generally appears. This may involve a significant divergencebetween the precoder parameters computed by the transmitters on theirown. The performance enhancement targeted by the cooperation istherefore not as high as expected, since the precoder parametersindependently determined by the transmitters involve residualinterference that grows with the CSIT mismatch.

It is desirable to overcome the aforementioned drawbacks of the priorart. It is more particularly desirable to provide a solution that allowsimproving performance of transmissions from a predefined set oftransmitters toward a predefined set of receivers in a MIMO-channelbased wireless communication system by relying on a precoder determinedin a distributed fashion among said transmitters, although CSIT mismatchmay exist.

SUMMARY OF INVENTION

To that end, the present invention concerns a method for performingtransmissions of data between a plurality of K_(t) transmitters and aplurality of K_(r) receivers via a global MIMO channel H=[H₁, . . .,H_(K) _(r) ] of a wireless communication system, by determining in adistributed fashion precoders to be applied for performing saidtransmissions, said precoders being respectively applied by saidtransmitters and jointly forming an overall precoder V, wherein each andevery j-th transmitter among said plurality of K_(t) transmittersperforms: gathering long-term statistics of Channel State Information atTransmitter CSIT errors incurred by each one of the K_(t) transmitterswith respect to the global MIMO channel H, the long-term statisticsdescribing the random variation of the CSIT errors; obtaining short-termCSIT related data and building its own view Ĥ^((j)) of the global MIMOchannel H; determining an estimate {tilde over (V)}^((j)) of the overallprecoder V from the obtained short-term CSIT related data; refining theestimate {tilde over (V)}^((j))[{tilde over (V)}₁ ^((j)), . . . ,{tildeover (V)}_(K) _(r) ^((j))] of the overall precoder Von the basis of thegathered long-term statistics of CSIT errors so as to obtain a refinedprecoder {tilde over (V)}^((j))[{tilde over (V)}₁ ^((j)), . . . ,{tildeover (V)}_(K) _(r) ^((j))] is a view of the overall precoder V from thestandpoint of said j-th transmitter, further on the basis of its ownview Ĥ^((j)) of the global MIMO channel H, and further on the basis of afigure of merit representative of performance of said transmissions viathe global MIMO channel H; and transmitting the data by applying aprecoder that is formed by a part of the refined precoder V^((j)) whichrelates to said j-th transmitter among said plurality of K_(t)transmitters.

Thus, performance of transmissions via the global MIMO channel of thewireless communication system is improved by relying on a precoderdetermined in a distributed fashion, although CSIT mismatch may exist.Robustness against CSIT mismatch is thus achieved without needing acentral unit to compute the precoder.

According to a particular feature, the figure of merit is a lower boundof a sum rate LBSR^((j)) reached via the global MIMO channel H, from thestandpoint of said j-th transmitter with respect to its own view Ĥ^((j))of the global MIMO channel H, as follows:

$\mspace{20mu} {{LBSR}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{\log {{\det \left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}^{- 1}}}}}$  whereinEMSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j))) = _({Δ⁽¹⁾, Δ⁽²⁾, …  , Δ^((K_(t)))|Ĥ^((j))})[MSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j)))]

wherein

represents the mathematical expectation and, wherein MSE_(k) ^((j)) (F₁^(j), . . . ,F_(K) _(r) ^((j))) represents mean square error matrixbetween the data to be transmitted and a corresponding filtered receivedvector for a realization of estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K)^(t) ⁾ which matches the long terms statistics of CSIT errors.

Thus, the sum of the rate of the receivers served by said transmissionsis improved by the determined precoder.

According to a particular feature, the figure of merit is the sum oftraces MINMSE^((j)), for k=1 to K_(r), of EMSE_(k) ^((j)) (F₁ ^(j), . .. ,F_(K) _(r) ^((j))) as follows:

$\mspace{20mu} {{MINMSE}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{{Trace}\left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}}}$  whereinEMSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j))) = _({Δ⁽¹⁾, Δ⁽²⁾, …  , Δ^((K_(t)))|Ĥ^((j))})[MSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j)))]

wherein E represents the mathematical expectation and, wherein MSE_(k)^((j)) (F₁ ^(j), . . . ,F_(K) _(r) ^((j))) represents the mean squareerror matrix between the data to be transmitted and a correspondingfiltered received vector for a realization of estimate errors Δ⁽¹⁾,Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ which matches the long terms statistics ofCSIT errors.

Thus, the average mean square error, as perceived by the receivers isimproved by the determined precoder.

According to a particular feature, refining the estimate {tilde over(V)}^((j)) of the overall precoder V is performed thanks to a refinementfunction f(. , .), as well as a set {F_(k) ^((j))} of refinementmatrices F_(k) ^((j)), k=1 to K_(r), in a multiplicative refinementstrategy, as follows:

V _(k) ^((j)) =f({tilde over (V)} _(k) ^((j)) ,{tilde over (F)} _(k)^((j)))={tilde over (V)} _(k) ^((j)) ,{tilde over (F)} _(k) ^((j))

Thus, such a multiplicative refinement allows for correcting CSITmismatch, especially in the context of block diagonalization precoding.

According to a particular feature, the overall precoder V is ablock-diagonalization precoder, the transmitters have cumulatively atleast as many antennas as the receivers, and refining the estimate{tilde over (V)}^((j)) of the overall precoder V thus consists inoptimizing the set {F_(k) ^((j))} of the refinement matrices F_(k)^((j)) with respect to the set {{tilde over (V)}_(k) ^((j))} of thematrices {tilde over (V)}_(k) ^((j)), which is obtained by applying aSingular Value Decomposition operation as follows:

Ĥ _([k]) ^((j)) =U _([k]) ^((j)) [D _([k]) ^((j)), 0][{tilde over (V)}′_([k]) ^((j)) ,{tilde over (V)}″ _(k) ^((j))]†

wherein Ĥ_([k]) ^((j)) represents a view of an aggregated interferencechannel estimation Ĥ_([k]) ^((j)) for the k-th receiver among the K_(r)receivers from the standpoint of said j-th transmitter, with

H _([k]) =[H† ₁ , . . . ,H† _(k−1) ,H† _(k+1) , . . . ,H† _(K) _(r) ]†

wherein {tilde over (V)}_(k) ^((j)) is obtained by selecting, accordingto a predefined selection rule similarly applied by any and alltransmitters, a predetermined set of N columns of the matrix {tilde over(V)}″_(k) ^((j)) resulting from the Singular Value Decompositionoperation, wherein each receiver has a quantity N of receive antennas.

Thus, a distributed block diagonal precoder is made robust to CSITmismatch.

According to a particular feature, the overall precoder V is aninterference aware coordinated beamforming precoder with block-diagonalshape, K_(t)=K_(r), and each transmitter has as a quantity M of transmitantennas equal to a quantity N of receive antennas of each receiver,each transmitter communicates only with a single receiver among theK_(r) receivers such that k=j,

wherein a sub-matrix W′_(k) such that:

V _(k) =E _(k) W′ _(k)

is computed as the eigenvector beamforming of the channel matrix definedby E_(k) ^(T)Ĥ^((k))E_(k), from a Singular Value Decomposition operationapplied onto, said channel matrix defined by E_(k) ^(T)Ĥ^((k))E_(k) asfollows:

E_(k) ^(T)Ĥ^((k))E_(k)=U′_(k)D′_(k)W′_(k)

wherein E_(k) is defined as follows:

E _(k)=[0_(M×(k−1)M) ,I _(M×M),0_(M×(K) _(t) _(−k)M)]^(T)

with 0_(M×(k−1)M) an M×(k−1)M sub-matrix containing only zeros, 0_(M×(K)_(t) _(−k)M) an M×(K_(t)−1)M sub-matrix containing only zeros, andI_(M×M) an M×M identity sub-matrix.

Thus, a distributed coordinated beamforming precoder is made robust toCSIT mismatch.

According to a particular feature, refining the estimate {tilde over(V)}^((j)) of the overall precoder V is performed thanks to a refinementfunction f(. , .), as well as a set {F_(k) ^((j))} of refinementmatrices F_(k) ^((j)), k=1 to K_(r), in an additive refinement strategy,as follows:

V _(k) ^((j)) =f({tilde over (V)} ^((j)) ,F _(k) ^((j)))={tilde over(V)} ^((j)) +F _(k) ^((j))

Thus, an additive refinement allows for correcting CSIT mismatch,especially in the context of regularized zeros forcing precoders.

According to a particular feature, the overall precoder V is aregularized zero-forcing precoder, and the estimate {tilde over(V)}^((j)) of the overall precoder V can be expressed as follows:

{tilde over (V)} ^((j))=(Ĥ ^((j))†Ĥ ^((j))+α^((j)) l)⁻¹ Ĥ _(k) ^((j))†

wherein α^((j)) is a scalar representing a regularization coefficientthat is optimized according to statistics of the own view Ĥ^((j)) of theglobal MIMO channel H from the standpoint of said j-th transmitter, andwherein α^((j)) is shared by said j-th transmitter with the othertransmitters among the K_(t) transmitters.

Thus, a Regularized Zero Forcing precoder is made robust to CSITmismatch.

According to a particular feature, refining the estimate {tilde over(V)}^((j)) of the overall precoder V is performed under the followingpower constraint:

Trace((f({tilde over (V)} ^((j)) ,F _(k) ^((j))))†f({tilde over (V)}^((j)) ,F _(k) ^((j))))=N

wherein each receiver has a quantity N of receive antennas.

Thus, transmission power is restrained.

The present invention also concerns a device for performingtransmissions of data between a plurality of K_(t) transmitters and aplurality of K_(r) receivers via a global MIMO channel H=[H₁, . . .,H_(K)] of a wireless communication system, by determining in adistributed fashion precoders to be applied for performing saidtransmissions, said precoders being respectively applied by saidtransmitters and jointly forming an overall precoder V, wherein saiddevice is each and every j-th transmitter among said plurality of K_(t)transmitters and comprises: means for gathering long-term statistics ofChannel State Information at Transmitter CSIT errors incurred by eachone of the k transmitters with respect to the global MIMO channel H, thelong-tem statistics describing the random variation of the CSIT errors;means for obtaining short-term CSIT related data and building its ownview Ĥ^((j)) of the global MIMO channel H; means for determining anestimate {tilde over (V)}^((j)) of the overall precoder V from theobtained short-term CSIT related data; means for refining the estimate{tilde over (V)}^((j))=[{tilde over (V)}₁ ^((j)), . . . ,{tilde over(V)}_(K) _(r) ^((j))] of the overall precoder Von the basis of thegathered long-term statistics of CSIT errors so as to obtain a refinedprecoder V^((j))=[V₁ ^((j)), . . . ,V_(K) _(r) ^((j))] that is a view ofthe overall precoder V from the standpoint of said j-th transmitter,further on the basis of its own view Ĥ^((j)) of the global MIMO channelH, and further on the basis of a figure of merit representative ofperformance of said transmissions via the global MEMO channel H; andmeans for transmitting the data by applying a precoder that is formed bya part of the refined precoder V^((j)) which relates to said j-thtransmitter among said plurality of K_(t) transmitters.

The present invention also concerns a computer program that can bedownloaded from a communications network and/or stored on a medium thatcan be read by a computer or processing device. This computer programcomprises instructions for causing implementation of the aforementionedmethod, when said program is run by a processor. The present inventionalso concerns an information storage medium, storing a computer programcomprising a set of instructions causing implementation of theaforementioned method, when the stored information is read from saidinformation storage medium and run by a processor.

Since the features and advantages related to the communications systemand to the computer program are identical to those already mentionedwith regard to the corresponding aforementioned method, they are notrepeated here.

The characteristics of the invention will emerge more clearly from areading of the following description of an example of embodiment, saiddescription being produced with reference to the accompanying drawings,among which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically represents a wireless communication system in whichthe present invention may be implemented.

FIG. 2 schematically represents an example of hardware architecture of acommunication device, as used in the wireless communication system.

FIG. 3 schematically represents an algorithm for determining, in adistributed fashion, precoders to be applied for transmitting data froma plurality of transmitters toward a plurality of receivers in thewireless communication system.

FIG. 4 schematically represents an iterative algorithm for determiningrefinement matrices used to refine the precoders.

DESCRIPTION OF EMBODIMENTS

FIG. 1 schematically represents a wireless communication system 100 inwhich the present invention may be implemented.

The wireless communication system 100 comprises a plurality oftransmitters, two 120 a, 120 b of which being represented in FIG. 1. Thewireless communication system 100 further comprises a plurality ofreceivers, two 110 a, 110 b of which being represented in FIG. 1. Forinstance, the transmitters 120 a, 120 b are access points or basestations of a wireless telecommunications network, and the receivers 110a, 110 b are mobile terminals having access to the wirelesstelecommunications network via said access points or base stations.

The transmitters 120 a, 120 b cooperate with each other in order toimprove performance when performing transmissions from the plurality oftransmitters 120 a, 120 b toward the plurality of receivers 110 a, 110 bvia wireless links 111 a, 111 b, 111 c, 111 d. The wireless link 111 arepresents the transmission channel from the transmitter 120 a to thereceiver 110 a, the wireless link 111 b represents the transmissionchannel from the transmitter 120 a to the receiver 110 b, the wirelesslink 111 c represents the transmission channel from the transmitter 120b to the receiver 110 a, and the wireless link 111 d represents thetransmission channel from the transmitter 120 b to the receiver 110 b.The transmitters 120 a, 120 b are interconnected, as shown by a link 121in FIG. 1A, so as to be able to exchange long-term statistics abouttransmission channel observations. The link 121 can be wired orwireless.

The cooperation is achieved by making the transmitters 120 a, 120 bapply respective precoders when performing said transmissions. Saidprecoders are determined in a distributed fashion within the wirelesscommunication system so that each transmitter determines the precoderthat said transmitter has to apply in the scope of said transmissions.More particularly, each transmitter (identified by an index j among theplurality of transmitters) determines, independently from the othertransmitters of said plurality, its own view V^((j)) of an overallprecoder V that should be cooperatively applied by said plurality oftransmitters for performing said transmissions. This aspect is detailedhereafter with respect to FIG. 3.

Herein the quantity of transmitters 120 a, 120 b in use is denotedK_(t), each transmitter having a quantity M of transmit antennas, andthe quantity of receivers 110 a, 110 b in use is denoted K_(r), eachreceiver having a quantity N of receive antennas. The receivers 110 a,110 b are configured to simultaneously receive signals from pluraltransmitters among the K_(t) transmitters. A global MIMO channel H isthus created between the K_(t) transmitters and the K_(r) receivers. Thepart of the global MIMO channel H which links a j-th transmitter amongthe K_(t) transmitters to a k-th receiver among the K_(r) receivers isrepresented by an N×M matrix herein denoted H_(k,j). One can note thatH_(k,j) is representative a MIMO channel too. The part of the globalMIMO channel H that links the k transmitters to the k-th receiver amongthe K_(r) receivers is a concatenation of the K_(t) MIMO channelsH_(k,j), with j=1 to K_(t), and is therefore an N×MK_(t) matrix hereindenoted H_(k). One can further note that H_(k) is representative of aMIMO channel too.

Let's consider a set of symbol vectors s_(k). Each symbol vector s_(k)of length N represents the data that has to be transmitted to the k-threceiver among the plurality of K_(r) receivers, at a given instant.Let's further denote s the stacked vector s=[s₁ ^(T),s₁ ^(T), . . .,s_(K) _(r) ^(T)]^(T) that contains all data to be transmitted by theK_(t) transmitters to the K_(r) receivers at said given instant, whereinA^(T) represents the transpose of a vector or matrix A.

Let's further consider the following overall precoder V:

V=[V₁, . . . ,V_(K) _(r) ]

and further define E_(j) ^(T)V, with j=1 to K_(t), as the part of theoverall precoder V to be applied by the j-th transmitter among the K_(t)transmitters, wherein E_(j) is an M×NK_(r) matrix such thatE_(j)=[0_(M×(j−1)M),I_(M×M),0_(M×(K) _(t) _(−j)M)]^(T), and whereinV_(k), with k=1 to K_(r), is the equivalent part of the overall precoderV which has to be applied to reach the k-th receiver among the K_(r)receivers. Inline with the notations already defined hereinbefore, oneshould note that V_(k) ^((j)) represents hereinafter the view of theprecoder equivalent part V_(k) from the standpoint of the j-thtransmitter among the K_(t) transmitters.

It should be noted that 0_(M×(j−1)M) in the expression of E_(j) aboverepresents an M×(j−1)M sub-matrix of E_(j) containing only zeros,0_(M×(j−1)M) represents an M×(K_(t)−j)M sub-matrix of E_(j) containingonly zeros, and I_(M×M) represents an M×M identity sub-matrix (could bean M×M identity matrix in other contexts herein).

In a joint processing CoMP (Coordinated Multipoint Transmission)approach, any and all transmitters know entirely the set of the symbolvectors s_(k) to be transmitted toward the K_(r) receivers at a giveninstant.

In a coordinated precoding approach where K_(t)=K_(r), each transmitteramong the K_(t) transmitters communicates with one receiver among theK_(r) receivers. It means that the j-th transmitter among the K_(t)transmitters only has to know the symbol vector s_(k) to be transmittedto the k-th receiver (with k=j) among the K_(r) receivers with whichsaid j-th transmitter communicates, which implies that the overallprecoder V has a block-diagonal shape. In the case where each and everyj-th transmitter among the K_(t) transmitters has to communicates withthe k-th receiver among the K_(r) receivers in such a way that k≠j,reordering of the K_(t) transmitters with respect to the index j and/orof the K_(r) receivers with respect to the index k is performed so as tomake the overall precoder V have a block-diagonal shape.

Considering the statements here above, a model of the wirelesscommunication system 100 can be expressed as follows:

$y_{k} = {{{H_{k}{Vs}} + n_{k}} = {{H_{k}V_{k}s_{k}} + {\sum\limits_{{ = 1},{ \neq k}}^{K_{r}}{H_{k}V_{}s_{}}} + n_{k}}}$

wherein:

-   -   y_(k) represents a symbol vector as received by the k-th        receiver among the K_(r) receivers when the symbol vector s_(k)        has been transmitted to said k-th receiver; and n_(k) represents        an additive noise incurred by said k-th receiver during the        transmission of the symbol vector s_(k).

It can be noticed that, in the formula above, the term H_(k)V_(k)s_(k)represents the useful signal from the standpoint of the k-th receiveramong the K_(r) receivers and the sum of the terms H_(k)

represents interference incurred by the k-th receiver among the K_(r)receivers during the transmission of the symbol vector s_(k).

A receive filter can be computed from the channel knowledge H_(k)V bythe k-th receiver among the K_(r) receivers, which may be obtained by adirect estimation if pilots precoded according to the overall precoder Vare sent by the concerned transmitter(s) among the K_(t) transmitters,or by obtaining the overall precoder V by signalling from the concernedtransmitter(s) among the K_(t) transmitters and further by estimatingthe MIMO channel H_(k) from pilots sent without precoding on this MIMOchannel H_(k).

When using a Zero-Forcing receive filter, the k-th receiver among theK_(r) receivers uses a linear filter T_(k) defined as follows:

T _(k)=((H _(k) V)†H _(k) V)⁻¹(H _(k) V)†

When using an MMSE receive filter, the k-th receiver among the K_(r)receivers uses a linear filter T_(k) defined as follows:

T _(k)=((H _(k) V)†H _(k) V1)⁻¹(H _(k) V)†

Then, a decision is made by said k-th receiver on the filtered receivedvector T_(k)y_(k) for estimating the symbol vector s_(k).

It has to be noted that, in the case where there is no effective receivefilter (for instance when Regularized Zero Forcing is applied by thetransmitters), T_(k)=1.

The K_(t) transmitters are configured to obtain CSIT (Channel StateInformation at the Transmitter). CSIT is obtained by each transmitteramong the k transmitters from:

-   -   feedback CSI (Channel State Information) from one more receivers        among the K_(r) receivers, and/or    -   channel estimation performed at said transmitter and using a        channel reciprocity property, and/or    -   from such CSI or such channel estimation provided by one or more        other transmitters among the K_(t) transmitters,    -   in such a way that CSI related data and/or channel estimation        related data propagation rules within the wireless communication        system 100 lead to CSIT errors and moreover to CSIT mismatch        among the K_(t) transmitters (i.e. different CSIT from the        respective standpoints of the K_(t) transmitters), for example        due to quantization operations.

One can note that, in addition to quantization operations, disparitiesin CSI related data effectively received by the K_(t) transmitters implythat differences in CSIT exist from one transmitter to another among theK_(t) transmitters, which leads to CSIT mismatch.

The global MIMO channel H can thus be expressed as follows, consideringeach and every j-th transmitter among the K_(t) transmitters:

H=Ĥ ^((j))+Δ^((j))

wherein Ĥ^((j)) represents a view of the global MIMO channel H from thestandpoint of the j-th transmitter among the K_(t) transmitters, whichis obtained by said j-th transmitter from the CSIT obtained by said j-thtransmitter, and wherein Δ^((j)) represents an estimate error betweenthe effective global MIMO channel H and said view Ĥ^((j)) of the globalMIMO channel H from the standpoint of said j-th transmitter. In asimilar manner, Ĥ_(k,i) ^((j)) denotes the view of the MIMO channelH_(k,i) from the standpoint of said j-th transmitter and Ĥ^((j)) denotesthe view of the MIMO channel H_(k) from the standpoint of said j-thtransmitter.

Therefore, the view V^((j)) of the overall precoder V might then beslightly different from one transmitter to another among the K_(t)transmitters, due to the CSIT mismatch. The j-th transmitter among theK_(t) transmitters then extracts, from the view V^((j)) of the overallprecoder V which has been determined by said j-th transmitter, theprecoder E_(j) ^(T)V^((j)) that said transmitter has to apply in thescope of said transmissions. As already mentioned, this is independentlyperformed by each transmitter among the K_(t) transmitters (j=1 toK_(t)). Optimization is therefore adequately performed so as to improvethe performance of the transmissions from the K_(t) transmitters to theK_(r) receivers, in spite of the CSIT mismatch and despite that eachj-th transmitter among the K_(t) transmitters independently determinesthe precoder E_(j) ^(T)V^((j)) that said transmitter has to apply. Thisaspect is detailed hereafter with regard to FIG. 3.

FIG. 2 schematically represents an example of hardware architecture of acommunication device, as used in the wireless communication system 100.The hardware architecture illustratively shown in FIG. 2 can representeach transmitter 120 a, 120 b of the wireless communication system 100and/or each receiver 110 a, 110 b of the wireless communication system100.

According to the shown architecture, the communication device comprisesthe following components interconnected by a communications bus 206: aprocessor, microprocessor, microcontroller or CPU (Central ProcessingUnit) 200; a RAM (Random-Access Memory) 201; a ROM (Read-Only Memory)202; an SD (Secure Digital) card reader 203, or an HDD (Hard Disk Drive)or any other device adapted to read information stored on a storagemedium; a first communication interface 204 and potentially a secondcommunication interface 205.

When the communication device is one receiver among the K_(r) receivers,the first communication interface 204 enables the communication deviceto receive data from the K_(t) transmitters via the global MIMO channelH. The second communication interface 205 is not necessary in this case.The first communication interface 204 further enables the communicationdevice to feed back channel state information to one or more transmitterdevices among the K_(t) transmitters.

When the communication device is one transmitter among the K_(t)transmitters, the first communication interface 204 enables thecommunication device to transmit data to the K_(r) receivers, via theglobal MIMO channel H, cooperatively with the other transmitters amongthe K_(t) transmitters. The first communication interface 204 furtherenables the communication device to receive channel state informationfed back by one or more receivers among the K_(r) receivers. Moreover,the second communication interface 205 enables the communication deviceto exchange data with one or more other transmitters among the K_(t)transmitters.

CPU 200 is capable of executing instructions loaded into RAM 201 fromROM 202 or from an external memory, such as an SD card. After thecommunication device has been powered on, CPU 200 is capable of readinginstructions from RAM 201 and executing these instructions. Theinstructions form one computer program that causes CPU 200 to performsome or all of the steps of the algorithm described herein.

Any and all steps of the algorithms described herein may be implementedin software by execution of a set of instructions or program by aprogrammable computing machine, such as a PC (Personal Computer), a DSP(Digital Signal Processor) or a microcontroller; or else implemented inhardware by a machine or a dedicated component, such as an FPGA(Field-Programmable Gate Array) or an ASIC (Application-SpecificIntegrated Circuit).

FIG. 3 schematically represents an algorithm for determining, in adistributed fashion within the wireless communication system 100,estimations of the overall precoder V to be applied for transmittingdata from the plurality of transmitters 120 a, 120 b toward theplurality of receivers 110 a, 110 b. The algorithm shown in FIG. 3 isindependently performed by each transmitter among the K_(t)transmitters. Let's illustratively consider that the algorithm of FIG. 3is performed by the transmitter 120 a, which is considered as the j-thtransmitter among the K_(t) transmitters.

In a first step S301, the transmitter 120 a gathers long-term statisticsabout the CSIT errors incurred by each one of the K_(t) transmitterswith respect to the global MIMO channel H. The long terms statisticsdescribe the random variation of the CSIT errors, which can for examplebe the variance of the CSIT errors.

By using a given statistical model of the CSIT errors, for example acentred Gaussian distribution, realizations of CSIT errors can begenerated from the gathered long-term statistics for simulating theimpact of said CSIT errors. Analytical derivation based on saidstatistical model and parameterized by said gathered long-termstatistics can be performed.

For instance, each j-th transmitter among the K_(t) transmittersestimates or computes a variance matrix Σ_(k,i) ^((j)) associated withthe channel estimation error between the MIMO channel estimation Ĥ_(k,i)^((j)) and the effective MIMO channel defined as follows: eachcoefficient of the variance matrix Σ_(k,i) ^((j)) is the variance of theerror between the corresponding coefficient of the MIMO channel matricesĤ_(k,i) ^((j)) and H_(k,i). It has to be noted that in this case thechannel estimation error is assumed to be independent from one channelcoefficient to another. In a variant, a covariance matrix of thevectorization of the difference (on a per-coefficient basis) Ĥ_(k,i)^((j))−H_(k,i) between the MIMO channel matrices Ĥ_(k,i) ^((j)) andH_(k,i) is estimated or computed.

When there is no exchange of CSIT information between the transmitters,Ĥ^((j)) represents an estimation, by the j-th transmitter among theK_(t) transmitters, of the global MIMO channel H. Said long termstatistics are representative of the error on the CSIT, which can becomputed according to a known behaviour divergence of the channelestimation technique in use with respect to the effective consideredMIMO channel and according to the effective CSI feedback from theconcerned receiver(s) to said j-th transmitter. For example, when pilotsymbols are sent in downlink for allowing each k-th receiver among theK_(r) receivers to estimate the MIMO channel H_(k), the resultingestimation error is proportional to the signal to noise plusinterference ratio via said MIMO channel H_(k), and the correspondingcoefficient of proportionality may be computed from the pilot density,such as in the document “Optimum pilot pattern for channel estimation inOFDM systems”, Ji-Woong Choi et al, in IEEE Transactions on WirelessCommunications, vol. 4, no. 5, pp. 2083-2088, Sept. 2005. This allowscomputing statistics relative to the downlink channel estimation error.When channel reciprocity is considered, each j-th transmitter among theK_(t) transmitters can learn the CSIT from a channel estimation inuplink direction, similar technique as in downlink is used to computethe uplink channel estimation error statistics. When a feedback link isused for obtaining the CSIT at the transmitter from a CSI feedbackcomputed from a channel estimation made by the concerned receiver(s),quantization error statistics on CSI can be estimated in the long termby the concerned receiver(s) and fed back to said j-th transmitter, orsaid quantization error statistics on CSI can be deduced from analyticalmodels. Indeed, each concerned receiver knows the effective CSI as wellas the quantization function, thus the effective quantization error.Said receiver is then able to compute the quantization error statisticsover time and is then able to feed them back to said j-th transmitter.For example, the receiver builds an histogram of the quantization errorrepresenting the distribution of the quantization error and feeds itback to the j-th transmitter. For example, the receiver and transmittersassume that the quantization error is multivariate Gaussian distributed,and the receiver estimates the mean vector and the covariance variancewhich are fed back to the j-th transmitter. Any of the above techniquescan be combined to obtain said CSIT error statistics associated to theestimation Ĥ^((j)) of the global MIMO channel H from the standpoint ofthe j-th transmitter. Then these long-term statistics can be exchangedbetween the transmitters, in such a way that each j-th transmitter amongthe K_(t) transmitters gathers long-term statistics about the CSITerrors incurred by each one of the K_(t) transmitters with respect tothe global MIMO channel H (which means that all the K_(t) transmittersshare the same long-term statistics).

In another example said long-term statistics are gathered as disclosedin the document “A cooperative channel estimation approach forcoordinated multipoint transmission networks”, Qianrui Li et al, IEEEInternational Conference on Communication Workshop (ICCW), pp. 94-99,8-12 Jun. 2015, where a combination of channel estimates is performedbetween transmitter nodes in order to compute the estimation Ĥ_(ki)^((j)) by each j-th transmitter among the K_(t) transmitters, of theMIMO channel H_(k,i), and the combination is then optimized to minimizethe mean square error associated with the difference (on aper-coefficient basis) Ĥ_(k,i) ^((j))−H_(k,i) between the MIMOchannel-matrices Ĥ_(k,i) ^((j)) and H_(k,i). The variance^(matrices Σ)_(k,i) ^((j)) ) are thus the result of the combination method describedin said document.

Thus, in a particular embodiment, the transmitter 120 a gathers thevariance matrices Σ_(k,i) ^((j)) which entries are the variance of theentries of the estimate error Δ_(k,i) ^((j)) between the effective MIMOchannel H_(k,i) and the estimation Ĥ_(k,i) ^((j)) of the MIMO channelH_(k,i) from the standpoint of the j-th transmitter among the K_(t)transmitters.

Once the step S301 has been performed by each one of the K_(t)transmitters, all the K_(t) transmitters share the same long-termstatistics about the CSIT errors. The step S301 is performed in anindependent process than the process typically in charge of effectivelyconfiguring the K_(t) transmitters so as to transmit the aforementionedset of the symbols vectors s_(k).

It can be noted that since the aforementioned statistics about the CSITerrors are by definition long-term data, the latency for ensuring thateach one of the K_(t) transmitters receives said long-term statisticshas low importance. Quantization is typically not a limiting factor fortransmitting such long-term statistics. On the contrary, the latency forpropagating data used by the K_(t) transmitters so that each transmitteramong the K_(t) transmitters is able to build its own CSIT is of mostimportance, in order for the wireless communication system 100 to havegood reactivity. Quantization with few levels can then be necessary fortransmitting such CIST related data and can drastically reduce thequality of the information. By the way, confusion shall be avoidedbetween long-term statistics about CSIT errors received by thetransmitter 120 a in the step S301 and CSIT related data needed by thetransmitter 120 a to have its own view Ĥ^((j)) of the global MIMOchannel H.

In a step S302, the transmitter 120 a obtains up-to-date (i.e.short-term) CSIT related data needed by the transmitter 120 a to haveits own view Ĥ^((j)) of the global MIMO channel H. The transmitter 120 apreferably shares the CSIT obtained in the step S302 with one or moretransmitters among the K_(t) transmitters, in order to help said one ormore transmitters to build their own respective view of the global MIMOchannel H.

Once the step S302 is performed by all the K_(t) transmittersindependently (substantially in parallel), the CSIT finally obtained bythe K_(t) transmitters differs from one transmitter to another one amongthe K_(t) transmitters, which leads to CSIT mismatch.

In a step S303, the transmitter 120 a determines an initial version{tilde over (V)}^((j)), from the standpoint of the transmitter 120 a(considered as the j-th transmitter among the K_(t) transmitters), ofthe overall precoder V from the CSIT related data obtained by thetransmitter 120 a in the step S302. The initial version {tilde over(V)}^((j)) of the overall precoder Vis therefore an estimate of theoverall precoder V. Since there is CSIT mismatch, this initial version{tilde over (V)}^((j)) of the overall precoder V may involve residualinterference that grows with the CSIT mismatch.

In a particular embodiment, the type of the overall precoder V and thusof the estimate {tilde over (V)}^((j)) of the overall precoder V areboth of one precoder type among the followings:

-   -   block-diagonalization precoders, for coordinated multi-point        transmissions with joint processing, as addressed in the        document “Cooperative Multi-Cell Block Diagonalization with        Per-Base-Station Power Constraints”, Rui Zhang, in IEEE Journal        on Selected Areas in Communications, vol. 28, no. 9, pp.        1435-1445, December 2010;    -   interference aware coordinated beamforming precoders, for        coordinated multi-point transmissions with coordinated        precoding, as addressed in the document “Interference        Aware-Coordinated Beamforming in a Multi-Cell System”,        Chan-Byoung Chae et al, in IEEE Transactions on Wireless        Communications, vol. 11, no. 10, pp. 3692-3703, October 2012;        and    -   regularized zero-forcing precoders, for coordinated multi-point        transmissions with joint processing, as addressed in the        document “A large system analysis of cooperative multicell        downlink system with imperfect CSIT”, Jun Zhang et al, in IEEE        International Conference on Communications (ICC), pp. 4813-4817,        10-15 Jun. 2012.

A particular embodiment of the present invention for each one of thesetypes of precoders is detailed hereafter.

It has to be noted that the initial version {tilde over (V)}^((j)) ofthe overall precoder V from the CSIT related data obtained by thetransmitter 120 a (considered as the j-th transmitter among the K_(t)transmitters) can be determined as indicated in the documents referencedabove with respect to each precoder type.

In a step S304, the transmitter 120 a refines the initial version {tildeover (V)}^((j)) of the overall precoder V according to the CSI errorlong-term statistics obtained in the step S301, so as to obtain arefined precoder V^((j))=[V₁ ^((j)), . . . ,V_(K) _(r) ^((j))], which isthe view of the overall precoder V from the standpoint of thetransmitter 120 a (considered as the j-th transmitter among the K_(t)transmitters).

In a particular embodiment, refining the initial version {tilde over(V)}^((j)) of the overall precoder V is performed by the transmitter 120a thanks to a refinement function f(. , .), as well as a set {F_(k)^((j))} of refinement matrices F_(k) ^((j)), k=1 to K_(r). Moreparticularly, considering that {tilde over (V)}^((j))=[{tilde over (V)}₁^((j)), . . . ,{tilde over (V)}_(K) _(r) ^((j))], refining the initialversion 17(j) of the overall precoder V means refining the sub-matrices{tilde over (V)}^((j)), k=1 to K_(r), of the initial version {tilde over(V)}^((j)) of the overall precoder V, wherein each sub-matrix {tildeover (V)}^((j)) is the equivalent within {tilde over (V)}^((j)) of thesub-matrix V_(k) ^((j)) within V^((j)).

Therefore, for each sub-matrix {tilde over (V)}_(k) ^((j)), therefinement function f (. , .) and the refinement matrix F_(k) ^((j)) canbe applied in a multiplicative refinement strategy, such as:

V _(k) ^((j)) =f({tilde over (V)} ^((j)) ,F _(k) ^((j)))={tilde over(V)} _(k) ^((j)) F _(k) ^((j))

or in an additive refinement strategy:

V _(k) ^((j)) =f({tilde over (V)} ^((j)) ,F _(k) ^((j)))={tilde over(V)} _(k) ^((j)) F _(k) ^((j))

preferably under the following power constraint:

Trace((f({tilde over (V)} ^((j)) ,F _(k) ^((j))))†f({tilde over (V)}^((j)) ,F _(k) ^((j))))=N

It has to be noticed from the relationships above that the size of eachrefinement matrix F_(k) ^((j)) depends on whether the refinementstrategy is additive or multiplicative.

Refining the initial version {tilde over (V)}^((j)) of the overallprecoder V is further performed as a function of the view Ĥ^((j)) of theglobal MIMO channel H from the standpoint of the transmitter 120 a(considered as the j-th transmitter among the K_(t) transmitters), aswell as of a figure of merit representative of performance oftransmissions from the transmitters to the receivers via the global MIMOchannel H, so as to be able to determine optimized version of the set{F_(k) ^((j))} of the refinement matrices F_(k) ^((j)), k=1 to K_(r). Insuch a wireless communication system, the figure of merit that isrepresentative of performance of the transmissions via the global MIMOchannel H is typically a multi-user performance metric.

It has to be noted that the precoder V^((j)) being a refined version ofthe initial version {tilde over (V)}^((j)) of the overall precoder Vfrom the standpoint of each j-th transmitter among the K_(t)transmitters computed from its own view Ĥ^((j)) of the global MIMOchannel H, a mismatch exists between the precoders V^((j)) independentlycomputed by all the K_(t) transmitters. Thus, a refinement operationshould be designed so as to minimize the impact of the mismatch on theperformance characterized by the figure of merit. It can be noted thatthe transmitters have two types of information for designing theprecoder V^((j)): first, the local CSIT, which is represented by theview Ĥ^((j)) of the global MIMO channel H from the standpoint of eachj-th transmitter, and which is exploited to compute the initial version{tilde over (V)}^((j)) of the overall precoder V, and the long termstatistics on estimate error between the effective global MIMO channel Hand said view Ĥ^((j)), which are shared between all transmitters and canthus be exploited for said refinement operation. Since astatistics-based refinement is considered, the refinement operation is astatistical method that computes a refined precoder V^((j)) out of a setof intermediate random variable {tilde over (V)}^((j)) characterizingthe possible overall precoder V in view of the previously determinedinitial version {tilde over (V)}^((j)) of the overall precoder V and ofthe long term statistics on estimate error between the effective globalMIMO channel H and said view Ĥ^((j)) for each j-th transmitter.Furthermore, the refinement strategy (multiplicative or additive) can bedefined in order to be able to statistically correct the initial version{tilde over (V)}^((j)) into V^((j)), said refinement strategy involvingparameters to be optimized so as to statistically reduce the impact ofthe mismatch on the performance.

Thus, each j-th transmitter can compute the distribution of anintermediate random variable {tilde over (V)}^((j)) (as definedhereafter), or generate realizations thereof, associated with theoverall precoder V after refinement by all the transmitters, accordingto the refinement strategy (multiplicative or additive) in use and tothe gathered long-term statistics about the CSIT errors, furtheraccording to the initial version {tilde over (V)}^((j)) of the overallprecoder V from the standpoint of said j-th transmitter and of it ownview Ĥ^((j)) of the global MIMO channel H, as detailed hereafter.

In a first particular embodiment, the figure of merit is a lower boundof a sum rate LBSR^((j)) reached via the global MIMO channel H, from thestandpoint of the transmitter 120 a (considered as the j-th transmitteramong the K_(t) transmitters), with respect to its own view Ĥ^((j)) ofthe global MIMO channel H. The sum rate lower bound LBSR^((j)) is afunction of the set {F_(k) ^((j))}. Considering that the transmitter 120a views the global MIMO channel H as being Ĥ^((j)), the sum rate lowerbound LBSR^((j)) is then defined as follows:

$\mspace{20mu} {{LBSR}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{\log {{\det \left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}^{- 1}}}}}$  whereinEMSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j))) = _({Δ⁽¹⁾, Δ⁽²⁾, …  , Δ^((K_(t)))|Ĥ^((j))})[MSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j)))]

wherein

represents the mathematical expectation and, wherein MSE_(k) ^((j))(F₁^((j)), . . . ,F_(K) _(r) ^((j))) represents the mean square errormatrix between the symbol vector s_(k) and the corresponding filteredreceived vector T_(k)y_(k) (as already explained) for a realization ofthe estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ which matches thelong terms statistics of CSIT error as obtained in the step S301, forexample by considering a centred Gaussian distribution of the CSITerrors, and for the view Ĥ^((j)) of the global MIMO channel H from thestandpoint of the transmitter 120 a. As explained in more detailshereafter, computing MSE_(k) ^((j)) for all and any k-th receiver amongthe K_(r) receivers would allow each considered j-th transmitter amongthe K_(t) transmitters to perform the optimization of the consideredfigure of merit. However, the effective realization of the estimateerrors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ is unknown. An approximation istherefore performed by relying on EMSE_(k) ^((j)) instead of MSE_(k)^((j)) thanks to the use of the long-term statistics about the CSITerrors, which have been gathered in the step S301.

The receiver filter T_(k) may be a function of the overall precoder Vand of the global MIMO channel H, which are unknown at the transmitters.But, each j-th transmitter can instead rely on its own view Ĥ^((j)) ofthe global MIMO channel H and realizations of the estimate errors Δ⁽¹⁾,Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ matching the long-term statistics gatheredin the step S301, so as to obtain the intermediate random variable{tilde over (V)}^((j))) and to obtain its own view T_(k) ^((j)) of eachreceive filter T_(k), k=1 to K_(r), as described hereafter. As a remark,the intermediate random variable {tilde over (V)}^((j)) and the viewT_(k) ^((j)) of the receive filter T_(k) are functions of the set tel ofthe refinement matrices F_(k) ^((j)), k=1 to K_(r).

Since EMSE_(k) ^((j)) (F₁ ^((j)), . . . ,F_(K) ^((j))) may be computedfor a fixed set of parameters F₁ ^((j)), . . ,F_(K) _(r) ^((j)), the sumrate lower bound LBSR^((j)) may be optimized by randomly definingseveral candidate sets of matrices to form the set {F_(k) ^((j))} of therefinement matrices F_(k) ^((j)), k=1 to K_(r), and keeping thecandidate set that minimizes the sum rate lower bound LBSR^((j)) fromthe standpoint of the transmitter 120 a.

In a preferred embodiment, an optimized sum rate lower bound LBSR^((j))is obtained thanks to an iterative algorithm as detailed hereafter withregard to FIG. 4.

In a second particular embodiment, the figure of merit is the sum oftraces, for k=1 to K_(r), of EMSE_(k) ^((j))(F₁ ^((j)), . . . ,F_(K)^((j))), as follows, which leads to a simplified expression MINMSE^((j))thus involving simpler implementation:

${MINMSE}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{{Trace}\left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}}$

Therefore, optimization of a figure of merit being a function ofEMSE_(k) ^((j)), such as the sum rate lower bound LBSR^((j)) or thesimplified expression MINMSE^((j)), according to the mathematicalexpectation of the realization of the estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . ., Δ^((K) ^(t) ⁾, matches the long terms statistics of CSIT error asobtained in the step S301, leads to obtaining the appropriate set {F_(k)^((j))} of the refinement matrices F_(k) ^((j), k=)1 to K_(r), and thento obtaining the refined precoder V^((j)) from said appropriate set{F_(k) ^((j))} of the refinement matrices F_(k) ^((j)) (by applying therelevant additive or multiplicative refinement strategy).

In a step S305, the transmitter 120 a performs, cooperatively with theother transmitters of the K_(t) transmitters, the transmission of theset of the symbol vectors s_(k) toward the K_(r) receivers. To do so,the transmitter 120 a applies the precoder E_(j) ^(T)V^((j)).

The algorithm of FIG. 3 may be applied independently by all thetransmitters on a regular basis. The algorithm of FIG. 3 may be appliedindependently by all the transmitters upon detecting that the globalMIMO channel H has changed beyond a predefined threshold. The algorithmof FIG. 3 may be applied independently by all the transmitters beforeeach transmission of data from the K_(t) transmitters toward the K_(r)receivers.

Particular embodiment for block-diagonalization precoders

In this particular embodiment, the overall precoder V is ablock-diagonalization precoder. It is then assumed that K_(t)M≥K_(r)N.By definition of block-diagonalization, considering the k-th receiveramong the K_(r) receivers, the interference induced by the MIMO channelsfor all other receivers among the K_(r) receivers is supposed to beeliminated, which means that, ideally:

H_(k)

=0, ∀

≠k

Let's denote H_([k)] an aggregated interference channel for the k-threceiver among the K_(r) receivers, which is expressed as follows:

H_([k)]=[H₁†, . . . ,H_(k−1)†,H_(k+1)†, . . . ,H_(K) _(r) †]†

and similarly Ĥ_([k)] an aggregated interference channel estimation forthe k-th receiver among the K_(r) receivers, which is expressed asfollows:

Ĥ_([k)]=[Ĥ₁†, . . . ,Ĥ_(k−1)†,Ĥ_(k+1)†, . . . ,Ĥ_(K) _(r) †]†

Applying a Singular Value Decomposition (SVD) operation on theexpression above of the aggregated interference channel H_([k)] resultsin:

H _([k]) =U _([k]) [D _([k]), 0][V′ _([k]) , V″ _(k)]†

wherein:

-   -   U_([k)] is an N(K_(r)−1)×N(K_(r)−1) unitary matrix,    -   D _([k)] is an N(K_(r)−1)×N(K_(r)−1) diagonal matrix,    -   V′_([k)] is an M K_(t)×N(K_(r)−1) matrix, and    -   V″_(k) is an MK_(t)×MK_(t)−N(K_(r)−1) matrix.        The size of the part V_(k) of the overall precoder V which has        to be applied for transmitting data toward the k-th receiver is        MK_(t)×N, and V_(k) is obtained by selecting a predetermined set        of N columns of the matrix V″_(k). The predetermined set can        either be, according to a predefined selection rule, the N first        columns of V″_(k) or the N last columns of V″_(k).

Considering the view H(J) of the global MIMO channel H from thestandpoint of the j-th transmitter among the K_(t) transmitters, theexpression above becomes:

Ĥ[k] ^((j)) U _([k]) ^((j)) [D _([k]) ^((j)),0][{tilde over (V)}′ _([k])^((j)) ,{tilde over (V)}″ _([k]) ^((j))]†

wherein Ĥ_([k]) ^((j)) represents the view of the aggregatedinterference channel estimation Ĥ_([k)] for the k-th receiver among theK_(r) receivers from the standpoint of the j-th transmitter among theK_(t) transmitters,wherein {tilde over (V)}′_([k]) ^((j)) is an MK_(t)×N(K_(r)−1) matrixequivalent to V′_([k)] when using the estimation Ĥ^((j)) instead of theeffective global MIMO channel H and {tilde over (V)}″_(k) ^((j)) is anMK_(t)×MK_(t)−N(K_(r)−1) matrix equivalent to V″_(k) when using theestimation Ĥ^((j)) instead of the effective global MIMO channel H,and wherein {tilde over (V)}_(k) ^((j)) is obtained by selecting apredetermined set of N columns of the matrix {tilde over (V)}″_(k)^((j)) according to the predefined selection rule, the selection rulebeing similarly applied by any and all transmitters, and wherein {tildeover (V)}_(k) ^((j)) is such that the refinement function f (. , .) isused in the aforementioned multiplicative refinement strategy, whichmeans:

V_(k) ^((j))={tilde over (V)}_(k) ^((j))F_(k) ^((j))

where F_(k) ^((j)) is a N×N matrix, preferably under the followingconstraint:

Trace(F _(k) ^((j)) F _(k) ^((j)))=N

As a result of the precoding strategy, the block-diagonalizationproperty is conserved, which means:

Ĥ_(k) ^((j))V_(l) ^((j))=0, ∀l≠k

However, it has to be noted that the block-diagonalization property isusually not achieved during the transmission on the global MIMO channelH, since a mismatch exists between Ĥ^((j)) and H. Thus, if thetransmitters use their initial version {tilde over (V)}^((j)) ofthe-precoder for-performing the transmissionsinterference exists betweenthe transmissions towards the receivers. This interference can bereduced by using the statistical knowledge on the long term statisticson estimate error between the effective global MIMO channel H and saidview Ĥ^((j)), by using the appropriate (multiplicative or additive)refinement strategy.

Therefore, by applying an SVD operation on the view Ĥ^((j)) of theglobal MIMO channel H from the standpoint of the transmitter 120 a(considered as the j-th transmitter among the K_(t) transmitters), thematrices {tilde over (V)}′[k_(]) ^((j)) and {tilde over (V)}_(k) ^((j))can be determined by the transmitter 120 a (considered as the j-thtransmitter among the K_(t) transmitters).

Refining the initial version {tilde over (V)}^((j)) of the overallprecoder V thus consists in optimizing the set {F_(k) ^((j))} of therefinement matrices F_(k) ^((j)) with respect to the set {{tilde over(V)}_(k) ^((j))} of the matrices {tilde over (V)}_(k) ^((j)) obtained bythe application of the Singular Value Decomposition on the view Ĥ^((j))of the global MIMO channel H from the standpoint of the transmitter 120a (considered as the j-th transmitter among the K_(t) transmitters).[0093]

First, a system performance metric is derived for a fixed realization ofthe estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ known by thetransmitters, and then a statistical analysis on the estimate errorsΔ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾, which are random variables, isapplied according to their respective long term statistics gathered atthe step S301.

Considering that an MMSE filter is implemented at each one of the K_(r)receivers for filtering signals received from the K_(t) transmitters,the expression of the MMSE filter, as computed at the k-th receiver fromthe perspective of the j-th transmitter and for a fixed realization ofthe estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ is:

$T_{k}^{(j)} = {\left( {\hat{V}}_{k}^{(j)} \right)^{\dagger}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)^{\dagger}\left( {{\sum\limits_{ = 1}^{K_{r}}{\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right){{\hat{V}}_{}^{(j)}\left( {\hat{V}}_{}^{(j)} \right)}^{\dagger}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)^{\dagger}}} + I} \right)^{- 1}}$

wherein {tilde over (V)}_(k) ^((j)) is the part of the intermediaterandom variable {tilde over (V)}^((j)) which concerns the k-th receiveramong the K_(r) receivers, by taking into account that the othertransmitters among the K_(t) transmitters also have performed arefinement according to a fixed realization of the estimate errors Δ⁽¹⁾,Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾.

Therefore, each j-th transmitter among the K_(t) transmitters computesthe following, for each and every

-th receiver among the K_(r) (

=1 to K_(r)):

${\hat{V}}_{}^{(j)} = \begin{bmatrix}{E_{1}^{\dagger}\left( {V_{}^{(j)} + {\left( {\hat{H}}_{\lbrack \rbrack}^{(j)} \right)^{+}\left( {\Delta_{\lbrack \rbrack}^{(1)} - \Delta_{}^{(j)}} \right)V_{}^{(j)}}} \right)} \\{E_{2}^{\dagger}\left( {V_{}^{(j)} + {\left( {\hat{H}}_{\lbrack \rbrack}^{(j)} \right)^{+}\left( {\Delta_{\lbrack \rbrack}^{(2)} - \Delta_{\lbrack \rbrack}^{(j)}} \right)V_{}^{(j)}}} \right)} \\\vdots \\{E_{K_{t}}^{\dagger}\left( {V_{}^{(j)} + {\left( {\hat{H}}_{\lbrack \rbrack}^{(j)} \right)^{+}\left( {\Delta_{\lbrack \rbrack}^{(K_{t})} - \Delta_{\lbrack \rbrack}^{(j)}} \right)V_{}^{(j)}}} \right)}\end{bmatrix}$

and wherein

represents the error estimation for the considered

-th receiver from the standpoint of the considered j-th transmitter, and

=[Δ₁ ^((j))†, . . . ,

, . . . , Δ_(K) _(r) ^((j))†]†

and

represents the view of the MIMO channel

from the standpoint of the considered j-th transmitter, and

represents an estimation of the aggregated interference channel

for the considered

receiver from the standpoint of the considered j-th transmitter.

Indeed, it is reminded that:

H=Ĥ ^((j))+Δ^((j))

which then means that, when a fixed realization of the estimate errorsΔ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ is known by the considered j-thtransmitter, said j-th transmitter can compute the view Ĥ^((j′)) of theglobal MIMO channel H from the standpoint of any other j′-th transmitteramong the K_(t) transmitters as follows:

Ĥ ^((j′)) =Ĥ ^((j))+Δ^((j))−Δ^((j′))

Thus computing

,

=1 to K_(r), as expressed above, allows then computing T_(k) ^((j)),which then allows defining MSE_(k) ^((j)) as follows:

$\begin{matrix}{{MSE}_{k}^{(j)} = {_{s_{k},n_{k}}\left\lbrack {\left( {s_{k} - {T_{k}^{(j)}y_{k}}} \right)\left( {s_{k} - {T_{k}^{(j)}y_{k}}} \right)^{\dagger}} \right\rbrack}} \\{= {{\left( {I - {{T_{k}^{(j)}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)}{\hat{V}}_{k}^{(j)}}} \right)\left( {I - {{T_{k}^{(j)}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)}{\hat{V}}_{k}^{(j)}}} \right)^{\dagger}} +}} \\{{{T_{k}^{(j)}\left( T_{k}^{(j)} \right)}^{\dagger} +}} \\{{{T_{k}^{(j)}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)}{\sum\limits_{ \neq k}{{{\hat{V}}_{}^{(j)}\left( {\hat{V}}_{}^{(j)} \right)}^{\dagger}\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)^{\dagger}\left( T_{k}^{(j)} \right)^{\dagger}}}}}\end{matrix}\quad$

wherein I is an identity matrix.

Since the effective fixed realization of the estimate errors Δ⁽¹⁾, Δ⁽²⁾,. . . , Δ^((K) ^(t) ⁾ is unknown at the transmitters in practice, butsince the estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ areassociated with a given probability of occurrence that can be computedfrom the long term statistics on CSIT error gathered in the step S301, astatistical analysis can be used.

Each j-th transmitter (such as the transmitter 120 a) is then able tocompute EMSK_(k) ^((j))(F₁ ^((j)), . . . ,F_(K) _(r) ^((j)) by using aMonte Carlo simulation, or by using a numerical integration, on thedistribution of Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ in view of the longterm statistics on CSIT error gathered in the step S301, further withrespect to the view Ĥ^((j)) of the global MIMO channel H from thestandpoint of the transmitter 120 a (considered as the j-th transmitteramong the K_(t) transmitters).

Alternatively a mathematical approximation provides a closed formexpression of EMSE_(k) ^((j))(F₁ ^((j)), . . . ,F_(K) _(r) ^((j))), asfollows:

$\mspace{20mu} \begin{matrix}{{{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} = \left( {I + {\left( {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{k}^{(j)}F_{k}^{(j)}} \right)^{\dagger} \times}} \right.} \\{\left( {{\sum\limits_{{ = 1},{ \neq k}}^{K_{r}}{{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}}} +} \right.} \\\left. {I + \Phi_{k}^{(j)}} \right)^{- 1} \\\left. {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{k}^{(j)}F_{k}^{(j)}} \right)^{- 1}\end{matrix}$   wherein$\Phi_{k}^{(j)} = {{\sum\limits_{ = 1}^{K_{r}}{\sum\limits_{\substack{t = 1 \\ t \neq j}}^{K_{t}}{{\hat{H}}_{k}^{(j)}E_{t}E_{t}^{\dagger}{{\hat{H}}_{\lbrack \rbrack}^{{(j)} +}\left\lbrack {{{\Theta_{\lbrack \rbrack}^{{(j)}{(t)}}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)}^{\dagger}E_{t}E_{t}^{\dagger}} + {{\Theta_{\lbrack \rbrack}^{{(j)}{(j)}}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)}^{\dagger}{\sum\limits_{\substack{{t\;}^{\prime} = 1 \\ {t\;}^{\prime} \neq j}}^{K_{t}}{E_{t^{\prime}}E_{t^{\prime}}^{\dagger}}}}} \right\rbrack}{\hat{H}}_{k}^{{(j)}\dagger}}}} + {{mdiag}\left( {\sum\limits_{{t = 1},{t \neq j}}^{K_{t}}{\Sigma_{k}^{(t)}E_{t}\mspace{14mu} {{diag}\left( {\sum\limits_{i = 1}^{K_{r}}{E_{t}^{\dagger}{{\hat{H}}_{\lbrack \rbrack}^{{(j)} +}\left( {\Theta_{\lbrack \rbrack}^{{(j)}{(t)}} + \Theta_{\lbrack \rbrack}^{{(j)}{(j)}}} \right)}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)^{\dagger}E_{t}}} \right)}}} \right)} + {{mdiag}\left( {\Sigma_{k}^{(j)}{{diag}\left( {\sum\limits_{ = 1}^{K_{r}}{{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}}} \right)}} \right)}}$  wherein$\mspace{20mu} {\Theta_{\lbrack \rbrack}^{{(j)}{(t)}} = {{mdiag}\left( {\Sigma_{\lbrack \rbrack}^{(t)}{{diag}\left( {{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}} \right)}} \right)}}$  and   Σ_(k)^((j)) = [Σ_(k, 1)^((j)), …  , Σ_(k, K_(t))^((j))]  and  Σ_([k])^((t)) = [(Σ₁^((t)))^(†), …  , (Σ_(k − 1)^((t)))^(†), (Σ_(k + 1)^((t)))^(†), …  , (Σ_(K_(r))^((t)))^(†)]^(†)

and wherein A⁺ is the Moore-Penrose pseudo-inverse of A, mdiag (.) makesa diagonal matrix from a given vector and diag (.) retrieves thediagonal entries of a matrix and stacks them into a vector.

Therefore, optimization of a figure of merit being a function ofEMSE_(k) ^((j)), such as the sum rate lower bound LBSRM or thesimplified expression MINMSE^((j)), according to the mathematicalexpectation of the realization of the estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . ., Δ^((K) ^(t) ⁾ which matches the long-term statistics of CSIT error asobtained in the step S301, leads to obtaining the appropriate set {F_(k)^((j))} of the refinement matrices F_(k) ^((j)), k=1 to K_(r), and thento obtaining the refined precoder V^((j)) from said appropriate set{F_(k) ^((j))} of the refinement matrices F_(k) ^((j)) by applying theaforementioned multiplicative refinement strategy.

In a preferred embodiment, an optimized sum rate lower bound LBSRU) isobtained thanks to the iterative algorithm as detailed hereafter withregard to FIG. 4.

Particular embodiment for interference aware coordinated beamformingprecoders

In this particular embodiment, it is assumed that the quantity K_(t) oftransmitters is equal to the quantity K_(r) of receivers, i.e.K_(t)=K_(r), and that the quantity M of transmit antennas is equal tothe quantity N of receive antennas, i.e. M=N. Moreover each one of theK_(t) transmitters communicates only with a single receiver among theK_(r). receivers. Considering any k-th receiver among the K_(r)receivers, the j-th transmitter among the K_(t) transmitters whichcommunicates with said k-th receiver is such that k=j. In themathematical expressions used hereafter in the particular embodiment forinterference aware coordinated beamforming precoders, the index k (onlyused hereinbefore for identifying any receiver among the K_(r)receivers) can be used instead of the index j. Interference awarecoordinated beamforming precoding is a sub-case of theblock-diagonalization precoding detailed above. Indeed, by consideringthat the overall precoder V has a block diagonal structure, with K_(t)blocks of size M, it is considered that each and every k-th transmitteramong the K_(t) transmitters only knows the symbol vector s_(k) (and notthe other symbol vectors s_(l), l≠k, that have to be transmitted by theother transmitters among the K_(t) transmitters), which is precoded byan M×M sub-matrix W′_(k) such that:

V_(k)=E_(k)W′_(k)

The sub-matrices W′_(k), for k=1 to K_(r), are obtained by implementingbeamforming and/or interference alignment based on the view Ĥ^((k)) ofthe global MIMO channel H from the standpoint of each and every k-thtransmitter. For example, the sub-matrices W′_(k) are computed accordingto an interference alignment technique, as in the document “DownlinkInterference Alignment” Changho Suh et al, IEEE Transactions onCommunications, vol. 59, no. 9, pp. 2616-2626, September 2011. Inanother example, the sub-matrices W′_(k) are computed as the eigenvectorbeamforming of the channel matrix defined by E_(k) ^(T)Ĥ^((k))E_(k),from an SVD operation applied onto said channel matrix by the consideredk-th transmitter among the K_(t) transmitters, such that:

E_(k) ^(T)Ĥ^((k))E_(k)=U′_(k)D′_(k)W′_(k)

wherein:

-   -   U′_(k) is an NK_(r)×NK_(r) unitary matrix, and    -   D′_(k) is an NK_(r)×NK_(r) diagonal matrix.

The optimization is then very similar to the approach described abovewith respect to the block-diagonalization precoding.

Then, the approach described above with respect to theblock-diagonalization precoding can thus be similarly applied, asfollows.

First, an initial version {tilde over (V)}^((j)) of the overall precoderV from the standpoint of each j-th transmitter among the K_(t)transmitters is computed from its own view Ĥ^((j)) of the global MIMOchannel H, such that the overall precoder V and the initial version{tilde over (V)}^((j)) thereof have a block diagonal structure. Eachblock defined by E_(k) ^(T){tilde over (V)}^((j))E_(k) is related to theprecoder used at the k-th transmitter from the standpoint of each j-thtransmitter, only for precoding the symbols vector s_(k), and is relatedto the sub-matrice W′_(k) previously described and determined accordingto an interference alignment or an the eigenvector beamformingtechnique.

The intial version {tilde over (V)}_(k) ^((j)) is such that therefinement function f (. , .) is used in the aforementionedmultiplicative refinement strategy, which means:

V_(k) ^((j))={tilde over (V)}_(k) ^((j))F_(k) ^((j))

where F_(k) ^((j)) is a N×N matrix, preferably under the followingconstraint:

Trace(F _(k) ^((j)) †F _(k) ^((j)))=N

Each j-th transmitter (such as the transmitter 120 a) is then able tocompute EMSE_(k) ^((j))(F₁ ^((j)), . . . ,F_(K) _(r) ^((j)))) by using aMonte Carlo simulation, or by using a numerical integration, on thedistribution of Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ which matches the longterms statistics of CSIT error as obtained in the step S301, furtherwith respect to the view Ĥ^((j)) of the global MIMO channel H from thestandpoint of the transmitter 120 a (considered as the j-th transmitteramong the K_(t) transmitters), under the constraint that the refinementmatrices F_(k) ^((j)) show block-diagonal respective shapes, by using

EMSE k ( j )  ( F 1 ( j ) , …  , F K r ( j ) ) = { Δ ( 1 ) , Δ ( 2 ) ,… , Δ ( K t )  H ^ ( j ) }  [ MSE k ( j )  ( F 1 ( j ) , …  , F K r( j ) ) ]

Alternatively a mathematical approximation provides a closed formexpression of EMSE_(k) ^((j))(F₁ ^((j)), . . . ,F_(K) _(r) ^((j))), asfollows:

$\mspace{20mu} \begin{matrix}{{{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} = \left( {I + {\left( {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{k}^{(j)}F_{k}^{(j)}} \right)^{\dagger} \times}} \right.} \\{\left( {{\sum\limits_{{ = 1},{ \neq k}}^{K_{r}}{{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}}} +} \right.} \\\left. {I + \Phi_{k}^{(j)}} \right)^{- 1} \\\left. {{\hat{H}}_{k}^{(j)}{\overset{\sim}{V}}_{k}^{(j)}F_{k}^{(j)}} \right)^{- 1}\end{matrix}$   wherein$\Phi_{k}^{(j)} = {{\sum\limits_{ = 1}^{K_{r}}{\sum\limits_{\substack{t = 1 \\ t \neq j}}^{K_{t}}{{\hat{H}}_{k}^{(j)}E_{t}E_{t}^{\dagger}{{\hat{H}}_{\lbrack \rbrack}^{{(j)} +}\left\lbrack {{{\Theta_{\lbrack \rbrack}^{{(j)}{(t)}}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)}^{\dagger}E_{t}E_{t}^{\dagger}} + {{\Theta_{\lbrack \rbrack}^{{(j)}{(j)}}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)}^{\dagger}{\sum\limits_{\substack{{t\;}^{\prime} = 1 \\ {t\;}^{\prime} \neq j}}^{K_{t}}{E_{t^{\prime}}E_{t^{\prime}}^{\dagger}}}}} \right\rbrack}{\hat{H}}_{k}^{{(j)}\dagger}}}} + {{mdiag}\left( {\sum\limits_{{t = 1},{t \neq j}}^{K_{t}}{\Sigma_{k}^{(t)}E_{t}^{\dagger}\mspace{14mu} {{diag}\left( {\sum\limits_{i = 1}^{K_{r}}{E_{t}^{\dagger}{{\hat{H}}_{\lbrack \rbrack}^{{(j)} +}\left( {\Theta_{\lbrack \rbrack}^{{(j)}{(t)}} + \Theta_{\lbrack \rbrack}^{{(j)}{(j)}}} \right)}\left( {\hat{H}}_{\lbrack \rbrack}^{{(j)} +} \right)^{\dagger}E_{t}}} \right)}}} \right)} + {{mdiag}\left( {\Sigma_{k}^{(j)}{{diag}\left( {\sum\limits_{ = 1}^{K_{r}}{{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}}} \right)}} \right)}}$  wherein$\mspace{20mu} {\Theta_{\lbrack \rbrack}^{{(j)}{(t)}} = {{mdiag}\left( {\Sigma_{\lbrack \rbrack}^{(t)}{{diag}\left( {{\overset{\sim}{V}}_{}^{(j)}{F_{}^{(j)}\left( {{\overset{\sim}{V}}_{}^{(j)}F_{}^{(j)}} \right)}^{\dagger}} \right)}} \right)}}$  and   Σ_(k)^((j)) = [Σ_(k, 1)^((j)), …  , Σ_(k, K_(t))^((j))]  and  Σ_([k])^((t)) = [(Σ₁^((t)))^(†), …  , (Σ_(k − 1)^((t)))^(†), (Σ_(k + 1)^((t)))^(†), …  , (Σ_(K_(r))^((t)))^(†)]^(†)

Therefore, optimization of a figure of merit being a function ofEMSE_(k) ^((j)), such as the sum rate lower bound LBSRW or M/NMSEW,according to the mathematical expectation of the realization of theestimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ which matches thelong-term statistics of CSIT error as obtained in the step S301, leadsto obtaining the appropriate set {F_(k) ^((j))} of the refinementmatrices F^(U), k=)1 to K_(r), and then to obtaining the refinedprecoder V^((j)) from said appropriate set {F_(k) ^((j))} of therefinement matrices F^((j)) by applying the aforementionedmultiplicative refinement strategy.

In a preferred embodiment, an optimized sum rate lower bound LBSR^((j))is obtained thanks to the iterative algorithm detailed hereafter withregard to FIG. 4.

Particular embodiment for regularized zero-forcing precoders

In this particular embodiment, the estimate {tilde over (V)}^((j)) ofthe overall precoder V from the standpoint of each and every j-thtransmitter among the K_(t) transmitters can be expressed as follows,with respect to each and every k-th receiver among the K_(r) receivers:

{tilde over (V)} _(k) ^((j))=(Ĥ ^((j))†Ĥ ^((j))+α^((j))1)⁻¹ Ĥ _(k)^((j))†

wherein α^((j)) is a scalar representing a regularization coefficientallowing to take into account a balance between interference and usefulsignal after channel inversion, and allowing optimizing theSignal-to-Interference-plus-Noise Ratio (SINR), and wherein α^((j)) isoptimized according to statistics of the view Ĥ^((j)) of the global MIMOchannel H from the standpoint of the considered j-th transmitter amongthe K_(t) transmitters, and wherein α^((j)) is shared by said j-thtransmitter with the other transmitters among the k transmitters, andwherein α^((j)) is obtained for example as in the document “RegularizedZero-Forcing for Multiantenna Broadcast Channels with User Selection”,Z. Wang et al, in IEEE Wireless Communications Letters, vol. 1, no. 2,pp. 129-132, April 2012, and wherein {tilde over (V)}_(k) ^((j)) is suchthat the refinement function f (. , .) is used in the aforementionedadditive refinement strategy, which means:

V _(k) ^((j)) i ={tilde over (V)} ^((j)) +F _(k) ^((j))

wherein F_(k) ^((j)) is a MK_(t)×N matrix.

First, a system performance metric is derived for a fixed realization ofthe estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ known by thetransmitters, and then a statistical analysis on the estimate errorsΔ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾, which are random variables, isapplied according to their respective long term statistics gathered atthe step S301.

{tilde over (V)}_(k) ^((j)) can be computed for a fixed realization ofΔ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ and with respect to the view Ĥ^((j))of the global MIMO channel H from the standpoint of the transmitter 120a (considered as the j-th transmitter among the K_(t) transmitters), asfollows:

$\begin{matrix}{{\hat{V}}_{k}^{(j)} = \begin{bmatrix}{E_{1}^{\dagger}V_{k}^{(1)}} \\\vdots \\{E_{K_{t}}^{\dagger}V_{k}^{(K_{t})}}\end{bmatrix}} \\{= \begin{bmatrix}{E_{1}^{\dagger}\left( {{\overset{\sim}{V}}_{k}^{(1)} + F_{k}^{(1)}} \right)} \\\vdots \\{E_{K_{t}}^{\dagger}\left( {{\overset{\sim}{V}}_{k}^{(K_{t})} + F_{k}^{(K_{t})}} \right)}\end{bmatrix}} \\{= \begin{bmatrix}{E_{1}^{\dagger}\left( {{\overset{\sim}{V}}_{k}^{(j)} + {C^{(j)^{- 1}}\left( {\Delta^{(1)} - \Delta^{(j)}} \right)} - {2{C^{{(j)}^{- 1}} \cdot {{Re}\left( {\left( {\Delta^{(1)} - \Delta^{(j)}} \right)^{\dagger}{\hat{H}}^{(j)}} \right)} \cdot {\overset{\sim}{V}}_{k}^{(j)}}} + F_{k}^{(1)}} \right)} \\{E_{K_{t}}^{\dagger}\left( {{\overset{\sim}{V}}_{k}^{(j)} + {C^{(j)^{- 1}}\left( {\Delta^{(1)} - \Delta^{(j)}} \right)} - {2{C^{{(j)}^{- 1}} \cdot {{Re}\left( {\left( {\Delta^{(1)} - \Delta^{(j)}} \right)^{\dagger}{\hat{H}}^{(j)}} \right)} \cdot {\overset{\sim}{V}}_{k}^{(j)}}} + F_{k}^{(K_{t})}} \right)}\end{bmatrix}}\end{matrix}\quad$

wherein Re{X} represents the real part of the complex input X,and wherein

C ^((j)) ⁻¹ =(Ĥ ^((j)) †Ĥ ^((j))+α^((j))1)⁻¹

which then allows defining MSE_(k) ^((j)) as follows:

${MSE}_{k}^{(j)} = \frac{1 + {\sum_{ \neq k}{{\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)^{\dagger}{\hat{V}}_{}^{(j)}}}^{2}}}{1 + {\sum_{}{{\left( {{\hat{H}}_{k}^{(j)} + \Delta_{k}^{(j)}} \right)^{\dagger}{\hat{V}}_{}^{(j)}}}^{2}}}$

It is reminded that EMSE_(k) ^((j)) is defined as follows:

EMSE k ( j )  ( F 1 ( j ) , …  , F K r ( j ) ) = { Δ ( 1 ) , Δ ( 2 ) ,… , Δ ( K t )  H ^ ( j ) }  [ MSE k ( j ) ]

The transmitter 120 a is then able to compute

EMSE_(k)^((j))(F₁^((j)), … , F_(K_(r))^((j)))

by using a Monte Carlo simulation, or by using a numerical integration,on the distribution of Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾ which matchesthe long terms statistics of CSIT error as obtained in the step S301,further with respect to the view Ĥ^((j)) of the global MIMO channel Hfrom the standpoint of the transmitter 120 a (considered as the j-thtransmitter among the K_(t) transmitters).

Therefore, optimization of a figure of merit being a function ofEMSE_(k) ^((j)), such as the sum rate lower bound LBSR^((j)) orMINMSE_(k) ^((j)), according to the mathematical expectation of therealization of the estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K) ^(t) ⁾which matches the long terms statistics of CSIT error as obtained in thestep S301, leads to obtaining the appropriate set {F_(k) ^((j))} of therefinement matrices F_(k) ^((j), k=)1 to K_(r), and then to obtainingthe refined precoder VU) from said appropriate {F_(k) ^((j))} of therefinement matrices F_(k) ^((j)) by applying the aforementioned additiverefinement strategy.

In a preferred embodiment, an optimized sum rate lower bound LBSR^((j))is obtained thanks to the iterative algorithm as detailed hereafter withregard to FIG. 4.

FIG. 4 schematically represents an iterative algorithm for determiningthe refinement matrices F_(k) ^((j)) from an optimization of LBSR(j) andthanks to the above descriptions on how to compute EMSE_(k) ^((j))(F₁^((j)), . . . ,F_(K) _(r) ^((j))). The algorithm of FIG. 4 is performedby each and every j-th transmitter among the K_(t) transmitters. Let'sillustratively consider that the algorithm of FIG. 4 is performed by thetransmitter 120 a, considered as the j-th transmitter among the K_(t)transmitters.

It is considered, when starting executing the algorithm of FIG. 4, thatthe transmitter 120 a knows the matrices {tilde over (V)}_(k) ^((j)),Σ_(k) ^((j)) and Ĥ_(k) ^((j)) for any and all k-th receivers among theK_(r) receivers.

In a step S401, the transmitter 120 a initializes the refinementmatrices F_(k) ^((j)), for each and every k-th receiver among the K_(r)receivers. The initialization can be set as random under the followingconstraint:

Trace((f({tilde over (V)} _(k) ^((j)) ,F _(k) ^((j))))†f({tilde over(V)} _(k) ^((j)) ,F _(k) ^((j))))=N

Alternatively, the refinement matrices F_(k) ^((j)) are taken asidentity N×N matrices for the block diagonal case, and MK_(t)×N matricescontaining only zeros for the regularized zero forcing case.

In a following step S402, the transmitter 120 a computes B_(k) ^((j)),for each and every k-th receiver among the K_(r) receivers, such that:

B _(k) ^((j))=EMSE_(k) ^((j))(F ₁ ^((j)) , . . . ,F _(K) _(r) ^((j)))

In a following step S403, the transmitter 120 a adjusts the refinementmatrices F_(k) ^((j)), for each and every k-th receiver among the K_(r)receivers, as follows:

$F_{1}^{(j)},\ldots \mspace{14mu},{F_{K_{r}}^{(j)} = {\underset{A_{1},\ldots \mspace{14mu},A_{K_{r}}}{\arg \mspace{14mu} \min}{\sum\limits_{i = 1}^{K_{r}}{{Trace}\mspace{11mu} \left( {B_{k}^{(j)} \cdot {{EMSE}_{k}^{(j)}\left( {A_{1},\ldots \mspace{14mu},A_{K_{r}}} \right)}} \right)}}}}$

such that the following constraint is preferably met:

Trace((f({tilde over (V)} _(k) ^((j)) ,F _(k) ^((j))))†f({tilde over(V)} _(k) ^((j)) ,F _(k) ^((j))))=N

In a following step S404, the transmitter 120 a checks whetherconvergence has been reached with respect to F_(i) ^((j)), . . . ,F_(K)_(r) ^((j)), for each and every k-th receiver among the K_(r) receivers.If such convergence has been reached, a step S405 is performed in whichthe algorithm of FIG. 4 ends; otherwise, the step 5402 is repeated, inwhich B_(k) ^((j)) is updated thanks to the values of F₁ ^((j)), . . .,F_(K) _(r) ^((j)) obtained in the last occurrence of the step S403.

The optimization of MINMSE^((j)) can also be done in order to determinethe refinement matrices F_(k) ^((j)) thanks to the above descriptions onhow to compute EMSE_(k) ^((j))(F₁ ^((j)), . . . ,F_(K) _(r) ^((j))).This leads to a convex optimization problem.

1-12. (canceled)
 13. A method for performing transmissions of databetween a plurality of K_(t) transmitters and a plurality of K_(r)receivers via a global MIMO channel H=[H₁, . . . ,H_(K) _(r) ] of awireless communication system, by determining in a distributed fashionprecoders to be applied for performing said transmissions, saidprecoders being respectively applied by said transmitters and jointlyforming an overall precoder V, wherein each and every j-th transmitteramong said plurality of K_(t) transmitters performs: obtainingshort-term CSIT related data and building its own view flu) of theglobal MIMO channel H; determining an estimate {tilde over (V)}^((j)) ofthe overall precoder V from the obtained short-term CSIT related data;wherein said j-th transmitter further performs: gathering long-termstatistics of Channel State Information at Transmitter CSIT errorsincurred by each one of the K_(t) transmitters with respect to theglobal MIMO channel H, the long-term statistics describing the randomvariation of the CSIT errors; refining the estimate {tilde over(V)}^((j))=[{tilde over (V)}₁ ^((j)), . . . ,{tilde over (V)}_(K) _(r)^((j))] of the overall precoder V on the basis of the gathered long-termstatistics of CSIT errors so as to obtain a refined precoder {tilde over(V)}^((j))=[{tilde over (V)}₁ ^((j)), . . . ,{tilde over (V)}_(K) _(r)^((j))] that is a view of the overall precoder V from the standpoint ofsaid j-th transmitter, further on the basis of its own view Ĥ^((j)) ofthe global MIMO channel H, and further on the basis of a figure of meritrepresentative of performance of said transmissions via the global MIMOchannel H; and transmitting the data by applying a precoder that isformed by a part of the refined precoder V^((j)) which relates to saidj-th transmitter among said plurality of K_(t) transmitters.
 14. Themethod according to claim 13, wherein the figure of merit is a lowerbound of a sum rate LBSR^((j)) reached via the global MIMO channel H,from the standpoint of said j-th transmitter with respect to its ownview Ĥ^((j)) of the global MIMO channel H, as follows:$\mspace{20mu} {{LBSR}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{\log {{\det \left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}^{- 1}}}}}$  whereinEMSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j))) = _({Δ⁽¹⁾, Δ⁽²⁾, …  , Δ^((K_(t)))|Ĥ^((j))})[MSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j)))]wherein F_(k) ^((j)), k=1 to K_(r) are refinement matrices, and wherein

represents the mathematical expectation and, wherein MSE_(k) ^((j))(F₁^((j)), . . . ,F_(K) _(r) ^((j))) represents mean square error matrixbetween the data to be transmitted and a corresponding filtered receivedvector for a realization of estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . , Δ^((K)^(t) ⁾ which matches the long terms statistics of CSIT errors.
 15. Themethod according to claim 13, wherein the figure of merit is the sum oftraces M/NMSE^((j)), for k=1 to K_(r), of MESE_(k) ^((j))(F₁ ^((j)), . .. ,F_(K) _(r) ^((j))), as follows:$\mspace{20mu} {{MINMSE}^{(j)} = {\sum\limits_{k = 1}^{K_{r}}{{Trace}\left( {{EMSE}_{k}^{(j)}\left( {F_{1}^{(j)},\ldots \mspace{14mu},F_{K_{r}}^{(j)}} \right)} \right)}}}$  whereinEMSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j))) = _({Δ⁽¹⁾, Δ⁽²⁾, …  , Δ^((K_(t)))|Ĥ^((j))})[MSE_(k)^((j))(F₁^((j)), …  , F_(K_(r))^((j)))]wherein Fe, k=1 to K_(r) are refinement matrices, and wherein

represents the mathematical expectation and, wherein MSE_(k) ^((j))(F₁^((j)), . . . ,F_(K) _(r) ^((j))) represents the mean square errormatrix between the data to be transmitted and a corresponding filteredreceived vector for a realization of estimate errors Δ⁽¹⁾, Δ⁽²⁾, . . . ,Δ^((K) ^(t) ⁾ which matches the long tenns statistics of CSIT errors.16. The method according to claim 13, wherein refining the estimate{tilde over (V)}^((j)) of the overall precoder V is performed thanks toa refinement function f (. , .), as well as a set {F_(k) ^((j))} ofrefinement matrices F_(k) ^((j)), k=1 to K_(r), in a multiplicativerefinement strategy, as follows:V _(k) ^((j)) =f({tilde over (V)}_(k) ^((j)) ,F _(k) ^((j)))={tilde over(V)}_(k) ^((j)) F _(k) ^((j))
 17. The method according to claim 16,wherein the overall precoder V is a block-diagonalization precoder, thetransmitters have cumulatively at least as many antennas as thereceivers, and in that refining the estimate {tilde over (V)}^((j)) ofthe overall precoder V thus consists in optimizing the set {F_(k)^((j))} of the refinement matrices F_(k) ^((j)) with respect to the set{{tilde over (V)}_(k) ^((j))} of the matrices {tilde over (V)}_(k)^((j)), which is obtained by applying a Singular Value Decompositionoperation as follows:Ĥ _([k]) ^((j)) =U _([k]) ^((j)) [D _([k]) ^((j)), 0][{tilde over (V)}′_([k]) ^((j)) ,{tilde over (V)}″ _(k) ^((j))]† wherein Ĥ_([k]) ^((j))represents a view of an aggregated interference channel estimationH_([k)] for the k-th receiver among the K_(r) receivers from thestandpoint of said j-th transmitter, withH _([k]) =[H† ₁ , . . . ,H† _(k−1) ,H† _(k+1) , . . . ,H† _(K) _(r) ]†wherein {tilde over (V)}_(k) ^((j)) is obtained by selecting, accordingto a predefined selection rule similarly applied by any and alltransmitters, a predetermined set of N columns of the matrix {tilde over(V)}″_(k) ^((j)) resulting from the Singular Value Decompositionoperation, wherein each receiver has a quantity N of receive antennas.18. The method according to claim 16, wherein the overall precoder V isan interference aware coordinated beamforming precoder withblock-diagonal shape, K_(t)=K_(r), and each transmitter has as aquantity M of transmit antennas equal to a quantity N of receiveantennas of each receiver, each transmitter communicates only with asingle receiver among the K_(r) receivers such that k=j, wherein asub-matrix W′_(k) such that:V_(k)=E_(k)W′_(k) is computed as the eigenvector beamforming of thechannel matrix defined by E_(k) ^(T) _(Ĥ) ^((k))E_(k), from a SingularValue Decomposition operation applied onto said channel matrix definedby E_(k) ^(T)Ĥ^((k))E_(k) as follows:E_(k) ^(T)Ĥ^((k))E_(k)=U′_(k)D′_(k)W′_(k) wherein E_(k) is defined asfollows:E _(k)=[0_(M×(k−1)M) , I _(M×M),0_(M×(K) _(t) _(−k)M)]^(T) with0_(M×(k−1)M) an M×(k−1)M sub-matrix containing only zeros, 0_(M×(K) _(t)_(−k)M) an M×(K_(t)−1)M sub-matrix containing only zeros, and I_(M×M) anM×M identity sub-matrix.
 19. The method according to claim 13, whereinrefining the estimate {tilde over (V)}^((j)) of the overall precoder Vis performed thanks to a refinement function f(. , .), as well as a set{F_(k) ^((j))} of refinement matrices F_(k) ^((j)), k=1 to K_(r), in anadditive refinement strategy, as follows:V _(k) ^((j)) =f({tilde over (V)} ^((j)) ,F _(k) ^((j)))={tilde over(V)} ^((j)) +F _(k) ^((j))
 20. The method according to claim 19, whereinthe overall precoder V is a regularized zero-forcing precoder, and theestimate {tilde over (V)}^((j)) of the overall precoder V can beexpressed as follows:{tilde over (V)} ^((j))=(Ĥ ^((j))†Ĥ ^((j))+α^((j)) l)⁻¹ Ĥ _(k) ^((j))†wherein α^((j)) is a scalar representing a regularization coefficientthat is optimized according to statistics of the own view Ĥ^((j)) of theglobal MIMO channel H from the standpoint of said j-th transmitter, andwherein α^((j)) is shared by said j-th transmitter with the othertransmitters among the K_(t) transmitters.
 21. The method according toclaim 13, wherein refining the estimate {tilde over (V)}^((j)) of theoverall precoder Vis perfoinied under the following power constraint:Trace((f({tilde over (V)} ^((j)) ,F _(k) ^((j))))†f({tilde over (V)}^((j)) ,F _(k) ^((j))))=N wherein f (. , .) is a refinement function andF_(k) ^((j)), k=1 to K_(r) are refinement matrices, and wherein eachreceiver has a quantity N of receive antennas.
 22. A non-transitorycomputer readable storage medium, wherein it stores a computer programcomprising program code instructions which can be loaded in aprogrammable device for implementing the method according to claim 13,when the program code instructions are run by the programmable device.23. A device for performing transmissions of data between a plurality ofK_(t) transmitters and a plurality of K_(r) receivers via a global MIMOchannel H=[H₁, . . . ,H_(K) _(r) ] of a wireless communication system,by determining in a distributed fashion precoders to be applied forperforming said transmissions, said precoders being respectively appliedby said transmitters and jointly forming an overall precoder V, whereinsaid device is each and every j-th transmitter among said plurality ofK_(t) transmitters and comprises a processor configured to: obtainshort-term CSIT related data and building its own view {tilde over(H)}^((j)) of the global MIMO channel H; determine an estimate {tildeover (V)}^((j)) of the overall precoder V from the obtained short-termCSIT related data; characterized in that said device further comprisesthe processor configured to: gather long-term statistics of ChannelState Information at Transmitter CSIT errors incurred by each one of theK_(t) transmitters with respect to the global MIMO channel H, thelong-term statistics describing the random variation of the CSIT errors;refine the estimate {tilde over (V)}^((j)) =[{tilde over (V)}₁ ^((j)), .. . ,{tilde over (V)}_(K) _(r) ^((j))] of the overall precoder V on thebasis of the gathered long-term statistics of CSIT errors so as toobtain a refined precoder {tilde over (V)}^((j)) =[{tilde over (V)}₁^((j)), . . . ,{tilde over (V)}_(K) _(r) ^((j))] that is a view of theoverall precoder V from the standpoint of said j-th transmitter, furtheron the basis of its own view Ĥ^((j)) of the global MIMO channel H, andfurther on the basis of a figure of merit representative of performanceof said transmissions via the global MIMO channel H; and transmit thedata by applying a precoder that is formed by a part of the refinedprecoder V^((j)) which relates to said j-th transmitter among saidplurality of K_(t) transmitters.